Equilibrium Pure States and Nonequilibrium Chaos

arXiv:cond-mat/9809375v1 [cond-mat.dis-nn] 28 Sep 1998

Equilibrium Pure States and Nonequilibrium Chaos C. M. Newman∗

D. L. Stein†

newman @ cims.nyu.edu

dls @ physics.arizona.edu

Courant Institute of Mathematical Sciences

Depts. of Physics and Mathematics

New York University

University of Arizona

New York, NY 10012, USA

Tucson, AZ 85721, USA

Abstract We consider nonequilibrium systems such as the Edwards-Anderson Ising spin glass at a temperature where, in equilibrium, there are presumed to be (two or many) broken symmetry pure states. Following a deep quench, we argue that as time t → ∞, although the system is usually in some pure state locally, either it never settles permanently on a fixed lengthscale into a single pure state, or it does but then the pure state depends on both the initial spin configuration and the realization of the stochastic dynamics. But this latter case can occur only if there exists an uncountable number of pure states (for each coupling realization) with almost every pair having zero overlap. In both cases, almost no initial spin configuration is in the basin of attraction of a single pure state; that is, the configuration space (resulting from a deep quench) is all boundary (except for a set of measure zero). We prove that the former case holds for deeply quenched 2D ferromagnets. Our results raise the possibility that even if more than one pure state exists for an infinite system, time averages don’t necessarily disagree with Boltzmann averages.

KEY WORDS: spin glass; nonequilibrium dynamics; deep quench; stochastic Ising model; broken ergodicity; coarsening; persistence; damage spreading.

1

Introduction

In this paper we study nonequilibrium dynamics of spin systems. While our approach is general, and covers both ordered and disordered, Ising and non-Ising systems, in the presence or absence of a magnetic field, we will for specificity here focus mostly on the dynamics of the Edwards-Anderson[12] Ising spin glass in zero field. We make no assumptions about the ∗

Partially supported by the National Science Foundation under grant DMS-95-00868 and DMS-98-02310. Partially supported by the U.S. Department of Energy under grant DE-FG03-93ER25155 and by the National Science Foundation under grant DMS-98-02153. †

1

real- or state-space structure of the low-temperature spin glass phase, but instead derive several general principles and then explore their consequences. Our results indicate that equilibrium pure state structure plays an important role in nonequilibrium dynamics. E.g., we will show that (at fixed temperature) a system with many pure states may have very different dynamical behavior than one with only a single pair1 — so nonequilibrium dynamics can serve as an important probe of the equilibrium pure state structure. That the number of and relationships among pure states can affect nonequilibrium dynamics may seem surprising in light of a general supposition (see, for example, Refs. [2, 29] and many of the references therein) that the time evolution of an infinite system is confined within a single pure state for all finite times. A consequence of this supposition is the prevalent notion that, if there exists at some temperature T more than one pure state (e.g., due to broken symmetry), then necessarily the limits t → ∞ (time) and N → ∞ (system size) do not commute (when, say, measuring the state of some observable). An equivalent statement is that time averages (as performed in the lab) and Boltzmann averages (as performed on paper or the computer) will give differing results. (One proposal[23] for avoiding this problem in theoretical treatments of spin glass dynamics is to start with a Boltzmann distribution of initial configurations, rather than a single one.) We will argue, however, that these are not necessary consequences of broken symmetry or multiple pure states; they may be true in some cases, but not in general. We examine here in which contexts they are true and in which they are not2 . We begin by constructing dynamical probability measures on spin configurations, which will enable us to clarify notions such as time evolution within a pure state.

2

Dynamical Measures

We will mostly, but not exclusively, consider the EA Ising spin glass in zero field. Its Hamiltonian is given by: X Jxy σx σy , (1) HJ = −

where the sites x, y ∈ Zd and the sum is taken over nearest neighbors only. The couplings Jxy are independent random variables, whose common probability density is symmetric about zero; we let J denote a particular realization of the couplings. 1

We emphasize that we are talking here about the equilibrium pure states present at the temperature at which the dynamics is observed, as distinguished from the proposal (see, e.g., Ref. [22]) that metastable states with O(1) free energy barriers affecting dynamics at a given temperature are the precursors of equilibrium pure states at lower temperatures 2 Except in some comments about other papers, we will not refer to metastable states, often proposed as responsible for the anomalous dynamical behavior of spin glasses. While we have no argument with this, we question the usefulness of the usual practice of inserting metastability by hand, requiring a guess as to the structure (usually in state space) and nature of the metastable states. In the treatment presented here, metastability may emerge naturally.

2

We consider Glauber dynamics of an infinite EA Ising spin glass starting from an initial (infinite-volume) spin configuration σ 0 . We regard σ 0 as chosen from the (infinite temperature) ensemble where the individual spins are independent random variables equally likely to be +1 or −1. This corresponds to the experimental situation following a deep quench, and also (for smaller systems) most numerical simulations. We denote by ω a given realization of the dynamics. So there are three sources of randomness in the problem, with realizations given by J , σ 0 , and ω. All three are needed to determine the spin configuration at time t (we take the starting time to be 0). From here on we take J to be fixed. We note that the exact choice of spin flip rates plays no role in our analysis at positive temperature, as long as detailed balance is satisfied. At T = 0 we consider only the widely used dynamical rule where flips that are energy lowering (or neutral or raising) occur at rate 1 (or 1/2 or 0). In all cases, ω can be regarded in the usual way as a collection of random times (tx,i : x ∈ Zd , i = 1, 2, . . .) when spin flips are considered (forming a Poisson process for each x) along with uniformly distributed random numbers ux,i that determine if the flips are taken. We now define a dynamical probability measure νt∗ ,τ (t∗ ) on the spin configurations, for 0 ≤ τ (t∗ ) ≤ t∗ . Given some σ 0 , we let the system evolve according to some ω up to a time t∗ − τ (t∗ ), after which we average over the dynamics up to time t∗ . That is, if we denote the dynamical (Markov) process as σ t = σ t (ω) for t ≥ 0, then νt∗ ,τ (t∗ ) is the conditional ∗ distribution of σ t conditioned on (J and σ 0 and) all (tx,i , ux,i)’s with tx,i ≤ t∗ − τ (t∗ ). So νt∗ ≡ νt∗ ,t∗ represents a complete averaging over the dynamics (and corresponds to the usual ∗ dynamical measure—i.e., to the distribution of σ t for fixed J and σ 0 ), while νt∗ ,0 represents no averaging at all (and hence a single spin configuration). To avoid awkward notation, we generally suppress the dependence, which is understood, of ν on J , σ 0 , ω (up to time t∗ − τ ), and T . We also note that neither t∗ nor τ depend on J , σ 0 or ω. In Section 5 we will briefly discuss the construction of measures based instead on time averaging for fixed dynamics. We can now begin to answer the question, what does it mean for the system to evolve or settle into (or “spend all its time inside”) a single pure state? Consider the cube ΛL of linear size L and volume |ΛL | (which may be arbitrarily large) centered at the origin. When T > 0 (but not at T = 0), we expect that for sufficiently large t∗ (depending on L) and for almost every σ 0 , the measure νt∗ approximates a (possibly mixed, possibly t∗ -dependent) Gibbs state restricted to the cube ΛL . By this we mean that there is some (infinite volume) Gibbs state ρt∗ such that for any L and any spin configuration σ (L) in ΛL , the probability assigned to σ (L) by the dynamical measure νt∗ and that assigned by the equilibrium Gibbs measure ρt∗ approach each other as t∗ → ∞; that is, νt∗ (σ (L) ) − ρt∗ (σ (L) ) → 0 as t∗ → ∞. More generally, we expect that for almost every σ 0 and ω, νt∗ ,τ (t∗ ) approximates some Gibbs state ρt∗ ,τ (t∗ ) providing only that τ (t∗ ) → ∞. The notion that, as t∗ increases, νt∗ ,τ (t∗ ) is increasingly well approximated by some infinite volume Gibbs state (possibly depending on t∗ ), may seem surprising — especially in view of frequent assertions (see, e.g., Ref. [9]) that equilibrium states are of little relevance for the nonequilibrium dynamics of infinite systems. In fact, this notion is nothing more than the property that for any L and any two spin configurations σ (L) , σ ′(L) , defined within ΛL and 3

agreeing on its (internal) boundary, the ratio νt∗ ,τ (σ (L) )/νt∗ ,τ (σ ′(L) ) converges to the usual finite volume Gibbs expression coming from the EA Hamiltonian. This kind of convergence may be expected as it is similar to the conjectured property of Glauber dynamics that only Gibbs states are stationary. To answer the question posed above about the meaning of settling into a single pure state, we now note that if τ (t∗ ) also grows sufficiently slowly, then νt∗ ,τ (t∗ ) should3 (for most ∗ ω’s) approximate a pure (i.e., extremal) Gibbs state ρα(t ) depending on σ 0 and ω (up to time t∗ − τ ). If on every fixed (and arbitrarily large) lengthscale L this ρα eventually becomes independent of time (after a timescale depending on L), then the system has settled into the pure state α. We now present our main results. Unless otherwise indicated, all are for T > 0 and are independent of space dimension. Our first result concerns the fully averaged dynamical measure νt∗ .

3

Evolution of the Dynamical Measure

Theorem 1. Given some J , assume that for almost every σ 0 , νt∗ converges to a limiting pure Gibbs state ν∞ as t∗ → ∞. Then ν∞ is the same pure state for almost every σ 0 . Proof. We use a coupling argument where a single ω is used with two starting spin realizations σ 0 and σ ′ 0 that differ at only finitely many sites. Then, by the nature of Glauber dynamics, there is a positive probability (with respect to the ω’s) that the two spin configurations merge after a finite time. Let θ > 0 represent the probability that the difference in configurations disappears by time t0 . So for t∗ ≥ t0 , 0

′0

µ′t∗ , µt∗ , νtσ∗ = θµt∗ + (1 − θ)˜ νtσ∗ = θµt∗ + (1 − θ)˜

(2)

˜ ′t∗ are some probability measures. It follows that for A any (measurable) ˜t∗ , and µ where µt∗ , µ set of spin configurations, ′0 0 (3) |νtσ∗ (A) − νtσ∗ (A)| ≤ 1 − θ . The same is true with t∗ replaced by ∞, first for A a locally defined event and then, by σ′0 σ0 approximation, for general A. The strict positivity of θ then implies that ν∞ and ν∞ cannot be mutually singular measures (i.e., living on completely disjoint regions of configuration space) and hence[20] , as pure states, they must be identical. So a change of finitely many σ0 spins in σ 0 doesn’t change ν∞ , and we can then use the Kolmogorov zero-one law[14] to conclude that the final pure state must be independent of σ 0 . ⋄ Despite the innocuous look of the theorem, it has important consequences. Its conclusion is obvious if there exists only one pure state, but it applies equally to the situation where 3

∗

The tracking of νt∗ ,τ (t∗) by the pure state ρα(t ) is in general weaker than the tracking by ρt∗ ,τ (t∗ ) in that α(t∗ ) may not have a limit and νt∗ ,τ (t∗ ) (σ (L) ) − ρα(t∗) (σ (L) ) may tend to zero only in probability rather ∗ than for almost all ω’s. We do expect, however, that for almost all ω’s, ρα(t ) tracks νt∗ ,τ (t∗ ) for most large t∗ ’s (i.e., when domain walls are far from the origin).

4

many pure states exist. Of course, it could logically be that only one pure state is “present” (in the sense that the conclusion of the theorem is valid) while other pure states exist but are not physically relevant in our dynamical context. (This might even be the case, for example, in the EA spin glass with a small nonzero field.) But if the conclusion of the theorem is not valid, then only one of two possibilities can occur4 : either (1) ν∞ is a mixed Gibbs state (which may or may not depend on σ 0 ), or (2) νt∗ does not converge as t∗ → ∞. We note that the latter case is already known to occur[17] in some 1D disordered ferromagnets at T = 0; on the other hand, if ν∞ exists and does not depend on σ 0 , then it would be analogous to (and perhaps the same as) ρJ , the average over the metastate discussed in Ref. [32]. Let’s consider each of these two possibilities, taking into account that νt∗ is the average over ω’s of νt∗ ,τ (t∗ ) . Possibility (1) implies that although νt∗ ,τ (t∗ ) (with properly chosen τ ) ∗ is approximately a pure state ρα(t ) , that pure state depends not only on σ 0 (as expected) but also on ω. This allows (but doesn’t require — see Subsection 4.1) the system always 0 to “land” in a pure state in the sense that νt∗ ,τ (t∗ ) → ρα(σ ,ω) — but then the pure state is (almost) never determined solely by σ 0 . Now, the basin of attraction of a pure state α ¯ may be defined as the set of σ 0 ’s such that α(σ 0 , ω) = α ¯ for almost every ω (see Ref. [13] for related discussions). We claim that if σ 0 is in the basin of attraction of some pure state, then by a modified version (see below) of the coupling argument used in the proof of Theorem 1, the same will be true after a change of finitely many spins in σ 0 and so, by the Kolmogorov zero-one law, the set of σ 0 ’s that are in some basin of attraction has probability either zero or one. Therefore, if many pure states are present (i.e., if α(σ 0 , ω) is not the same for almost all σ 0 and ω), then the union of all their basins of attraction must form a set of measure zero in the space of σ 0 ’s; i.e., the configuration space resulting from a deep quench is all “boundary” in the sense that almost every initial configuration will land in one of several (or many) pure states depending on the realization of the dynamics (if it lands at all). The modified coupling argument is as follows. Let σ 0 and σ ′ 0 be fixed and let D denote the finite set of x’s where they differ. Rather than using the same ω for the coupled processes, we take an ω = (tx,i , ux,i) and an ω ′ = (t′x,i, u′x,i) that are identical for x outside D but for x inside D, they are identical only for times after σ t and σ ′ t merge. For earlier times, ω and ω ′ inside D are independent of each other. If A′ is an event defined in terms only of ω ′ with Prob(A′ ) > 0 and M denotes the event of eventual merger of the two processes, then since ω inside D may (with small but strictly positive probability) force a merger by a small time ε, it follows that Prob(A′ ∩ M) > 0. Assuming α(σ 0 , ω) = α(σ ¯ 0 ) for almost every ω, we take A′ to be the complement of the event that α(σ ′ 0 , ω ′) = α(σ ¯ 0 ) so that Prob(A′ ∩ M) > 0 would 0 0 yield the contradiction that Prob(α(σ , ω) = α(σ ¯ )) < 1. It follows that Prob(A′ ) = 0 and so α(σ ′ 0 , ω ′) = α(σ ¯ 0 ) for almost every ω ′, which is exactly the claim made in the previous paragraph. It has been speculated[27] that slow relaxation in spin glasses may be due to points in (high-dimensional) state space always being “near” a boundary. What we’ve shown here 4

Logically, there is a third possibility that ν∞ is not a Gibbs state, but this would violate the expected behavior discussed in Section 2 and hence we disregard it (except when T = 0 — see Section 4 and Ref. [30]).

5

differs in fundamental respects: our conclusion is that almost every point in state space is actually on a boundary, and therefore the dynamical consequences are not restricted to very low temperatures. We note finally that Theorem 1 may be relevant to damage spreading[1, 21, 24] , where one asks whether the damage (i.e., discrepancy) between σ t and σ ′ t (with a single ω) grows as t → ∞. Theorem 1 suggests that if damage spreading occurs, then νt∗ doesn’t converge to a single pure state (e.g., it might converge to a mixed state, as above).

4

Local Non-Equilibration

Before discussing possibility (2), let us consider the physical picture implied by Theorem 1. Roughly speaking, some time after an initial quench the system will form domains, whose average size increases with time, corresponding to the different pure states. This scenario has been analyzed for the two-state droplet picture[15, 25, 34] . It is also a well-known scenario for coarsening in a ferromagnet following a deep quench[6] . (Of course, in contrast to the spin glass case, one does know how to prepare a ferromagnet in a pure state; for a general discussion, see Ref. [33].) In Ref. [6] it was stated that the infinite homogeneous ferromagnet never reaches equilibrium in any finite time (following a deep quench) because the domain sizes (in this case, of positive and negative magnetization) increase with time but are never infinite on any finite timescale. We do not consider this by itself to be nonequilibration because it does not preclude the possibility that on any finite lengthscale, the system equilibrates after some finite time, in the sense that after that time domain walls cease to move across the region. Instead, we now propose a much stronger version of nonequilibration — the possibility of local non-equilibration (LNE) on any finite lengthscale, which is implied by possibility (2) and could also occur with possibility (1). We will discuss the difference between these two cases of LNE in Subsection 4.1, but for now will not distinguish between the two. We will also discuss below the relation between LNE and persistence exponents[7, 10, 11, 28] . By LNE we mean that in any fixed finite region, the system never settles down into a pure state. That is, domain walls do not simply move farther from the region as time progresses, but continually return and sweep across it, changing the state within. More precisely, LNE is said to occur unless there is some choice of τ (t∗ ) such that for almost all σ 0 0 and ω, νt∗ ,τ (t∗ ) → ρα(σ ,ω) , a pure state. If LNE occurs, it would force us to revise the usual dynamical definition[2] of, e.g., the EA order parameter. It could also mean that, for infinite systems, time averages and Gibbs averages could agree, despite the presence of many pure states. We will return to this in Section 5 after investigating LNE in more detail by means of the next theorem, which applies to both homogeneous and disordered systems. Theorem 2. If only a single pair (or countably many, including a countable infinity) of pure states exists (with fixed J ) and these all have nonzero EA order parameter, then LNE occurs. Proof. Suppose that there exists at T (and for the given J ) only a single pair of pure 6

states, and assume that LNE does not occur so that for each σ 0 and ω, there is a limiting pure state α(σ 0 , ω). The overlap of α(σ 0 , ω) and α(σ ′0 , ω ′ ) is Q(J , σ 0 , ω, σ ′0 , ω ′) = lim |ΛL |−1 L→∞

X

< σx >α(σ0 ,ω) < σx >α(σ′0 ,ω′ ) ,

(4)

x∈ΛL

where < · >α denotes the (thermal) average with respect to the pure state ρα . The possible overlap values can be only ±qEA and both of these outcomes must have positive probability of occurring (as σ 0 , ω, σ ′0, ω ′ vary independently to yield a pair of replicas). But the translationergodicity of each of the distributions from which J , σ 0 , ω, σ ′0, ω ′ are chosen implies the same for the joint (product) distribution of (J , σ 0 , ω, σ ′0 , ω ′). Thus, the fact that the overlap Q is a translation-invariant function implies that Q must be constant for almost all realizations, leading to a contradiction. This argument can be immediately extended to any countable number of pure state pairs. ⋄ This proof also shows that if LNE does not occur (and the limiting pure states α(σ 0 , ω) have nonzero qEA ), then almost every (as σ 0 , ω, σ ′0 , ω ′ vary) overlap of the pair of pure states, α(σ 0 , ω) and α(σ ′0 , ω ′ ), is zero. This leads to: Corollary. If LNE does not occur and qEA 6= 0, then there must be an uncountable number of pure states, with almost every pair (in the above sense) having overlap zero. This shows that, as stated in the introduction, nonequilibrium dynamics provides important information on the structure of equilibrium pure states. It also suggests a dynamical test of the two-state picture: search for chaotic time dependence in νt∗ ,τ (t∗ ) . If LNE does not occur, then the two-state picture has been ruled out: there must be an uncountable number of states, almost all of which have overlap zero (consistent with the results of Ref. [32]). If LNE does occur then neither the two-state nor the many-state pictures have been ruled out. How might one go about observing LNE in a spin glass, where, unlike the ferromagnet, one doesn’t know what a domain wall looks like? Here one can use the clustering property that characterizes pure states in general. E.g., a truncated 2-point correlation function of the form < σx σ0 > − < σx >< σ0 > approaches 0 as |x| → ∞ if the averaging is done in a pure state, but not otherwise. In the current context a pure state average corresponds to a dynamical average using νt∗ ,τ with τ ≪ t∗ . In principle one could evaluate this correlation function numerically for t∗ ≫ τ ≫ |x| ≫ 1; if it does not approach zero for increasing values of these parameters, that would constitute a clear signal of the occurrence of LNE. An important consequence of Theorem 2 is that LNE must also occur at positive temperature in the 2D uniform Ising ferromagnet and the random Ising ferromagnet for d < 4. (However, the argument for LNE in random ferromagnets is not entirely rigorous as there is no complete analysis of interface pure states there[18] . There is though a rigorous proof that for the SOS approximation, these states do exist for d ≥ 4 and do not for d < 4; see Refs. [4, 5]). Moreover, in the 2D homogeneous ferromagnet (on the square lattice) results on LNE can be extended to T = 0, in the sense there that σ t does not converge as t → ∞; in fact, every spin flips infinitely often. The proof (by contradiction) is based on showing that if 7

some spin (say at the origin) remained fixed forever, then (by translation ergodicity and spin flip symmetry) so would two spins of opposite sign on the x and y axes. But then there would be a domain wall passing through the rectangle determined by these three spins and “cutting off” one of them. In every time interval there would be a nonzero probability of the domain wall moving to flip that spin, and so with probability one, it eventually would flip. For the full proof, see Ref. [30]. It is also shown in Ref. [30] that for many systems σ t does converge to some limit at T = 0. This is based on a very general argument that there can be only finitely many flips at any site that strictly lower the energy. Examples include spin glasses and random ferromagnets where the common distribution of the Jxy ’s is continuous, and either has a finite mean (as in most ordinary models) or else is sufficiently “spread out”, as in the highly disordered spin glass or ferromagnet[31] . Other examples are homogeneous ferromagnets (or homogeneous antiferromagnets or ±J spin glasses) on lattices with an odd number of nearest neighbors, such as the 2D hexagonal lattice or a double-layered 2D square lattice. (There are also systems, such as the ±J spin glass on the square lattice, where some spins flip only finitely many times and some spins flip infinitely often[19] .) In light of these results, we restrict the term LNE to T > 0, since in the zero-temperature situations where σ t converges, the limit configuration is typically only metastable rather than a ground state and so equilibration has not really occurred. In these systems one can define a dynamical order parameter, related to the autocorrelation, that does not decay to zero. To further clarify the discussion of LNE for the ferromagnet, we note that it is a phenomenon separate from the spontaneous formation (at positive temperature) of domains of, say, the minus phase within the plus phase. That is, on a timescale exponential in L, there will form a domain containing the origin (of size L) of the minus phase. Similarly, for a finite system of size L, the entire system will flip back and forth between the plus and minus phases on a timescale exponential in L. However, this is different from LNE, which presumably takes place on time scales of some power of L. Also, the spontaneous formation of droplets described above cannot occur at T = 0, but as already discussed, in the 2D ferromagnet the phenomenon of domain walls forever sweeping across any finite region persists at zero temperature. Since the existence of LNE for all T < Tc in the 2D Ising ferromagnet may seem surprising, we present a possible physical mechanism for this case which may also shed light on LNE in general. The initial spin configuration has (with probability one) no infinite domains. As the configuration evolves, some domains shrink and others coalesce. So the origin should always be contained in a finite domain, whose size will usually be slowly decreasing, but sporadically will have a large change either by coalescing or because a domain wall passes through the origin and the identity of the domain changes. Thus LNE is primarily the result of nonequilibrium domain wall motion (driven by mean curvature) combined with the complex domain structure resulting from the original quench. It is also consistent with phase separation (as would be expected from equilibrium roughening arguments). In particular, the occurrence of LNE does not preclude the divergence with t of the mean scale of the domain containing the origin (although for fixed σ 0 and ω, there is a much more complex 8

behavior, as indicated above).

4.1

LNE and chaotic time dependence

We noted earlier in this section that LNE can occur in the context of either possibility (1) (the fully averaged dynamical measure νt∗ has a limit, which is a mixed state) or (2) (νt∗ does not converge). That is, LNE must occur if possibility (2) holds, but may or may not occur if possibility (1) holds. We now explore further the distinctions between the two cases of LNE. As described earlier, one way for possibility (1) to occur is if, for some choice of τ (t∗ ), 0 0 νt∗ ,τ (t∗ ) → ρα(σ ,ω) for almost all σ 0 and ω, where ρα(σ ,ω) is some pure state. But it could also happen that for any choice of τ (t∗ )